It is often required to determine the rate of rotation of a moving body using apparatus which is both highly accurate as well as being sensitive to the measured parameter whilst being relatively insensitive to other influences.
It is known to employ gyroscopes for the determination of rate of rotation. The use of a gyroscope for such measurement resides in the principle of spinning a symmetrical rotor at very high speed about its axis of symmetry. Consequently, there will exist a very high angular momentum about this axis and, according to Law of Conservation of Angular Momentum, the angular momentum of the rotor about the spinning axis will tend to be conserved in the event of an external rotation applied to the gyroscope.
Thus, as an external rotation is applied to the gyroscope, a compensating moment is applied thereto whose magnitude is a function of the applied rate of spin. In reality, angular momentum is not exactly conserved on account of frictional and other losses. Therefore, in order to achieve good results, frictional losses must be minimized and the angular momentum of the rotor should be made as large as possible. Therefore, in order for gyroscopes to be sufficiently sensitive, it is necessary for a relatively massive rotor to be spun within substantially frictionless bearings at a very high rate of spin.
Such systems are inherently expensive and subject the bearings to very high forces. This, in turn, imposes a relatively short lifespan on the gyroscope.
Consequently, in spite of the popularity of the gyroscope for measuring rates of rotation, there have been moves in recent years to employ the Coriolis effect in so-called "non-gyroscopic" inertia measuring devices. The principle of the Coriolis effect is that when a body moves linearly in a specified direction whilst, at the same time, being subjected to a rotation about an axis perpendicular to the direction of linear motion, then the linear and angular velocities combine vectorially to produce a force which is applied to the body in a direction which is mutually perpendicular both to the spin axis and the direction of linear motion. The magnitude of the resultant force, called the Coriolis force, is a function of the rate of rotation at which the body rotates and may therefore be used as a basis for its determination. Thus, if:
.omega.=the angular velocity vector of the body, PA0 v=the linear velocity vector of the body, PA0 m=the mass of the body, and PA0 F.sub.c =the magnitude of the Coriolis force, then EQU F.sub.c =2m.omega..times.v (1) PA0 .omega..sub.z =rate of rotation about the z-axis, PA0 y.sub.O =amplitude of forced periodic vibration, PA0 .OMEGA.=the frequency of the forced periodic vibration, and PA0 .xi.=damping ratio
where .omega..times.v is the vector cross product of the vectors .omega. and v.
The Gyrotron utilizes this phenomenon by employing a tuning-fork type of element rotated about its longitudinal axis. The tines of the fork are subjected to a forced high frequency oscillation by means of a pair of electromagnetic drive coils. Since the forced oscillation is perpendicular to the axis of rotation of the fork, a Coriolis force will be generated along a mutually perpendicular, transverse axis, the magnitude of which force is detected by means of a pair of electromagnetic pick-up coils. Determination of the Coriolis response may be used to determine the rate of rotation of the fork about its longitudinal axis.
The basic principle of the Gyrotron described above has been exploited in many prior art devices for determining rate of rotation. For example, British Published Patent Specification No. 2 154 739 discloses a gyroscopic device having a disc-shaped piezo-electric resonator along a surface of which are provided a plurality of exciting electrodes interspersed with a like plurality of detecting electrodes. When the piezoelectric resonator is spun about its longitudinal axis and a sinusoidal exciting voltage is applied to the exciting electrodes, there is generated, in accordance with the Coriolis principle described above, a voltage signal at the detecting electrodes which are disposed 90.degree. out of phase with the exciting electrodes. On account of the rotation of the disc, there exists a phase shift in the electrical output from the detecting electrodes and this phase shift is employed within a feedback loop in order to null the voltage difference between the electrical output of the two pairs of detecting electrodes. Under these circumstances, the voltage derived across one pair of the detecting electrodes provides a direct measure of the angular velocity .omega. of the rotating disc.
U.S. Pat. No. 3,839,915 discloses a turn rate sensor of the vibratory tuning fork type, as described above with respect to the Gyrotron. In such an arrangement, a rotation about an axis parallel to the tines of the fork in combination with forced vibration of the tines themselves, gives rise to a Coriolis force along a mutually perpendicular transverse axis. The system further provides for the compensation of asymmetry of the tuning fork and misalignment of the tine motions, so as to minimize errors.
Likewise, in U.S. Pat. No. 4,930,351 (Macy et al.) there is disclosed a "Tuning Fork" type angular rate sensor operating in accordance with the principles of the Gyrotron described above. Macy et al. disclose a multi-sensor comprising two parts. The arrangement shown in FIG. 1 thereof serves to measure displacements of the tuning fork tines each of which is analogous to a single degree of freedom system. The sensing is based on two beams permitting accelerating sensing owing to the axial movement which results from out-of-plane asymmetrical deflection of the beams caused by one of the beams being provided with a thin, flexible portion so as to render it asymmetric.
U.S. Pat. No. 4,884,446 (Ljung) discloses a force balance wherein a proof mass is restrained within the X-Y plane. The proof masses are balanced longitudinally and their centers of gravity are located such that the latter do not move when the proof masses are angularly vibrated. Consequently, there is no coupling between the translational and angular vibrations of the proof masses. The translational vibration having a spring constant k.sub.1 =12EIl.sup.-3 serves to isolate beam vibration. Angular rate sensing is based on the angular forced vibration of the inertias with the angular spring constant k.sub.2 =EIl.sup.-1 about the center of mass of each inertia in the mode of a simple one degree of freedom angular vibration.
Furthermore, the system disclosed by Ljung is a first order resonant system configured to have a high Q-value resulting from the absence of damping gas and nodal suspensions. The high Q-value is also due to the minimization of energy losses resulting from the fact that the ends of the inertia members are attached to the beam ends such that the entire beam length is evenly flexurally stressed with the inertia members vibrating about their centers of gravity.
As is known, first order resonant systems are highly sensitive when operating at the resonant frequency, but suffer from the major drawback that any small deviation from the resonant frequency results in a marked decrease in operational sensitivity.
This drawback is associated with all the systems described above which are first order mechanical resonant systems having two degrees of freedom. Thus, if the three mutually perpendicular Cartesian axes are considered, the systems rotate about the vertical z-axis and it is the rate of rotation about this axis which is to be determined. A forced vibration is applied along the transverse y-axis giving rise to a Coriolis response along the transverse x-axis. In the particular case wherein the magnitude of the forced vibration along the y-axis is constrained to be constant, one of the degrees of freedom is lost and the resulting system has one degree of freedom only. However, since the basic, unconstrained system has two degrees of freedom, it will be referred to in hereinafter as a two degree of freedom system.
It may be shown that for such mechanically resonant systems having two degrees of freedom and wherein:
then, the response in the direction of the x-axis due to the Coriolis effect is given by: ##EQU1##
Several drawbacks associated with prior art constructions become clear on an analysis of the complete system. Thus, the response X is highly sensitive to the resonant frequency .omega..sub.n. It may be shown that if the damping ratio, .xi., equals 0.02 and the frequency .OMEGA. of the forced periodic vibration is only 1% less than the resonant frequency .omega..sub.n, then the magnitude of the response X falls by as much as 20%.
There thus exists a conflict between the desire, on the one hand, to raise the gain (i.e. the magnitude of X) and, on the other hand, to be insensitive to the fixed parameters of the mechanical system.
Furthermore, if the mass rotates sinusoidally according to the equation: EQU .omega..sub.z =.omega..sub.zo cos .lambda.t (3)
then the bandwidth of the system is given by: EQU .lambda..sub.B.W. =.xi..OMEGA. (4)
Thus, to increase the bandwidth of .lambda. the product .xi..OMEGA. must be increased. However, it will be seen from equation (2) above that increasing the product .xi..OMEGA. decreases the magnitude of the response X. In other words, by increasing the bandwidth, the sensitivity is lowered.
In addition to the drawbacks with respect to the frequency response of first order systems having a single degree of freedom, as described above, there are additional problems associated with coupling between the forced vibration and the response, owing to the nature of the system.
In conclusion, there exists an inherent problem in employing the model disclosed in the prior art having a single degree of freedom, since such systems are highly sensitive to the accuracy at which the frequency .OMEGA. of the forced vibration approximates the resonant frequency .omega..sub.n of the system. To maximize the gain, the mass must be vibrated at the natural frequency, i.e. .OMEGA.=.omega..sub.n. If, however, for any reason there is even a small discrepancy between .OMEGA. and .omega..sub.n, then the gain drops drastically.